4. each package contains 10 marbles. which graph correctly represents the relationship between the number of packages and the number of marbles? (a) graph a (b) graph b (c) graph c (d) graph d 5. make a table to represent the number of packages and the number of marbles. 6. which table below represents a proportional relationship? (a) input 1 2 3 4 5 output 5 6 7 8 9 (b) input 1 2 3 4 5 output 6 7 8 9 10 (c) input 1 2 3 4 5 output 4 3 2 1 0 (d) input 1 2 3 4 5 output 5 10 15 20 25 7. does the graph below show a proportional relationship? explain how you know.

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Accepted Answer

### Question 4
# Explanation:
## Step1: Determine the relationship
Each package has 10 marbles, so the number of marbles $ y = 10x $ (where $ x $ is packages). For $ x = 1 $, $ y = 10 $; $ x = 2 $, $ y = 20 $; etc.
## Step2: Analyze Graphs
- Graph A: At $ x = 1 $, $ y = 10 $; $ x = 2 $, $ y = 20 $; $ x = 3 $, $ y = 30 $; $ x = 4 $, $ y = 40 $; $ x = 5 $, $ y = 50 $; $ x = 6 $, $ y = 60 $. Matches $ y = 10x $.
- Graph B: Doesn't follow $ y = 10x $ (e.g., $ x = 1 $, $ y = 20 $ which is not 10).
- Graph C: Points are not linear with $ y = 10x $.
- Graph D: Doesn't follow $ y = 10x $ (e.g., $ x = 2 $, $ y = 10 $ which is not 20).
# Answer: (a) Graph A

### Question 5
# Explanation:
## Step1: Define the relationship
Marbles = 10 × Packages (since each package has 10 marbles).
## Step2: Fill the table
- Packages = 0: Marbles = $ 10×0 = 0 $
- Packages = 1: Marbles = $ 10×1 = 10 $
- Packages = 2: Marbles = $ 10×2 = 20 $
- Packages = 3: Marbles = $ 10×3 = 30 $
- Packages = 4: Marbles = $ 10×4 = 40 $
- Packages = 5: Marbles = $ 10×5 = 50 $
# Answer:
| Packages | Marbles |
|----------|---------|
| 0        | 0       |
| 1        | 10      |
| 2        | 20      |
| 3        | 30      |
| 4        | 40      |
| 5        | 50      |

### Question 6
# Explanation:
## Step1: Proportional Relationship Definition
A proportional relationship has $ \text{Output} = k×\text{Input} $ (constant $ k $, and passes through (0,0) if input 0 is considered).
## Step2: Analyze each option
- (a): Output - Input = 4 (not proportional, it's linear with $ k = 1 $, but output starts at 5 when input 1, so $ 5 = 1×1 + 4 $, not proportional).
- (b): Output - Input = 5 (linear, not proportional as $ 6 = 1×1 + 5 $, not $ k×1 $ with $ k = 6 $ for input 1).
- (c): Output = -Input + 5 (linear, not proportional).
- (d): Output = 5×Input (5 = 5×1, 10 = 5×2, 15 = 5×3, 20 = 5×4, 25 = 5×5). Constant $ k = 5 $, so proportional.
# Answer: (d) Input 1 2 3 4 5; Output 5 10 15 20 25

### Question 7
# Explanation:
## Step1: Proportional Relationship Check
A proportional relationship is a straight line through the origin ($ (0,0) $) and has constant $ \frac{y}{x} $.
## Step2: Analyze the Graph
- The graph has points (3,1), (4,2), (5,3), (6,4), (6,5)? Wait, looking at the axes: x is Peter’s Age, y is Jane’s Age. The line does not pass through (0,0) (when Peter’s age is 0, Jane’s age isn’t 0). Also, $ \frac{y}{x} $ is not constant (e.g., $ x = 3 $, $ y = 1 $ ( $ \frac{1}{3} $ ); $ x = 4 $, $ y = 2 $ ( $ \frac{2}{4} = \frac{1}{2} $ ); not constant).
## Step3: Conclusion
Since it doesn't pass through the origin and $ \frac{y}{x} $ isn't constant, it's not proportional.
# Answer: No. A proportional relationship must be a straight line through the origin (0,0) and have a constant ratio $ \frac{y}{x} $. This graph does not pass through (0,0) (when Peter’s age is 0, Jane’s age is not 0) and the ratio of Jane’s age to Peter’s age is not constant (e.g., at $ x = 3 $, $ \frac{y}{x}=\frac{1}{3} $; at $ x = 4 $, $ \frac{y}{x}=\frac{2}{4}=\frac{1}{2} $, which is not constant), so it is not a proportional relationship.

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QUESTION IMAGE

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Explanation:

Question 4

Step1: Determine the relationship

Step2: Analyze Graphs

Step1: Define the relationship

Step2: Fill the table

Step1: Proportional Relationship Definition

Step2: Analyze each option

Answer:

Question 5